A* is similar to Dijkstra with reduced cost
What you are doing when calculating $d'(x,y)$:


$d(x,y)$: calculating the original edge distance from $x$ to $y$
$h(y)$: plus the heuristic from $y$ to the goal
$h(x)$: minus the heuristic from $x$ to the goal


So, using this recalculation of the original edge-values ($1.$) in Dijkstra's algorithm you are inherently accounting for the heuristic component of A* by incorporating it ($2.$) into the value of the edge traversed, and discarding ($3.$) the accumulated previous heuristic values of previous nodes in the path.

The additional condition $h(x) ≤ d(x, y) + h(y)$ ensures the new edge-values are strictly positive.